Copied to
clipboard

## G = C3⋊S3×C3⋊C8order 432 = 24·33

### Direct product of C3⋊S3 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C3⋊S3×C3⋊C8
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C32×C3⋊C8 — C3⋊S3×C3⋊C8
 Lower central C33 — C3⋊S3×C3⋊C8
 Upper central C1 — C4

Generators and relations for C3⋊S3×C3⋊C8
G = < a,b,c,d,e | a3=b3=c2=d3=e8=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 600 in 164 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, C2×C3⋊C8, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C324C8, C3×C24, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, S3×C3⋊C8, C8×C3⋊S3, C32×C3⋊C8, C337C8, C12×C3⋊S3, C3⋊S3×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊S3, C3⋊C8, C4×S3, C2×Dic3, S32, C2×C3⋊S3, S3×C8, C2×C3⋊C8, S3×Dic3, C4×C3⋊S3, S3×C3⋊S3, S3×C3⋊C8, C8×C3⋊S3, Dic3×C3⋊S3, C3⋊S3×C3⋊C8

Smallest permutation representation of C3⋊S3×C3⋊C8
On 144 points
Generators in S144
(1 36 61)(2 37 62)(3 38 63)(4 39 64)(5 40 57)(6 33 58)(7 34 59)(8 35 60)(9 121 118)(10 122 119)(11 123 120)(12 124 113)(13 125 114)(14 126 115)(15 127 116)(16 128 117)(17 77 30)(18 78 31)(19 79 32)(20 80 25)(21 73 26)(22 74 27)(23 75 28)(24 76 29)(41 130 98)(42 131 99)(43 132 100)(44 133 101)(45 134 102)(46 135 103)(47 136 104)(48 129 97)(49 85 90)(50 86 91)(51 87 92)(52 88 93)(53 81 94)(54 82 95)(55 83 96)(56 84 89)(65 105 141)(66 106 142)(67 107 143)(68 108 144)(69 109 137)(70 110 138)(71 111 139)(72 112 140)
(1 53 141)(2 54 142)(3 55 143)(4 56 144)(5 49 137)(6 50 138)(7 51 139)(8 52 140)(9 24 129)(10 17 130)(11 18 131)(12 19 132)(13 20 133)(14 21 134)(15 22 135)(16 23 136)(25 44 114)(26 45 115)(27 46 116)(28 47 117)(29 48 118)(30 41 119)(31 42 120)(32 43 113)(33 86 70)(34 87 71)(35 88 72)(36 81 65)(37 82 66)(38 83 67)(39 84 68)(40 85 69)(57 90 109)(58 91 110)(59 92 111)(60 93 112)(61 94 105)(62 95 106)(63 96 107)(64 89 108)(73 102 126)(74 103 127)(75 104 128)(76 97 121)(77 98 122)(78 99 123)(79 100 124)(80 101 125)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(41 68)(42 69)(43 70)(44 71)(45 72)(46 65)(47 66)(48 67)(57 78)(58 79)(59 80)(60 73)(61 74)(62 75)(63 76)(64 77)(81 116)(82 117)(83 118)(84 119)(85 120)(86 113)(87 114)(88 115)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 121)(97 107)(98 108)(99 109)(100 110)(101 111)(102 112)(103 105)(104 106)(129 143)(130 144)(131 137)(132 138)(133 139)(134 140)(135 141)(136 142)
(1 36 61)(2 62 37)(3 38 63)(4 64 39)(5 40 57)(6 58 33)(7 34 59)(8 60 35)(9 118 121)(10 122 119)(11 120 123)(12 124 113)(13 114 125)(14 126 115)(15 116 127)(16 128 117)(17 77 30)(18 31 78)(19 79 32)(20 25 80)(21 73 26)(22 27 74)(23 75 28)(24 29 76)(41 130 98)(42 99 131)(43 132 100)(44 101 133)(45 134 102)(46 103 135)(47 136 104)(48 97 129)(49 85 90)(50 91 86)(51 87 92)(52 93 88)(53 81 94)(54 95 82)(55 83 96)(56 89 84)(65 105 141)(66 142 106)(67 107 143)(68 144 108)(69 109 137)(70 138 110)(71 111 139)(72 140 112)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)

G:=sub<Sym(144)| (1,36,61)(2,37,62)(3,38,63)(4,39,64)(5,40,57)(6,33,58)(7,34,59)(8,35,60)(9,121,118)(10,122,119)(11,123,120)(12,124,113)(13,125,114)(14,126,115)(15,127,116)(16,128,117)(17,77,30)(18,78,31)(19,79,32)(20,80,25)(21,73,26)(22,74,27)(23,75,28)(24,76,29)(41,130,98)(42,131,99)(43,132,100)(44,133,101)(45,134,102)(46,135,103)(47,136,104)(48,129,97)(49,85,90)(50,86,91)(51,87,92)(52,88,93)(53,81,94)(54,82,95)(55,83,96)(56,84,89)(65,105,141)(66,106,142)(67,107,143)(68,108,144)(69,109,137)(70,110,138)(71,111,139)(72,112,140), (1,53,141)(2,54,142)(3,55,143)(4,56,144)(5,49,137)(6,50,138)(7,51,139)(8,52,140)(9,24,129)(10,17,130)(11,18,131)(12,19,132)(13,20,133)(14,21,134)(15,22,135)(16,23,136)(25,44,114)(26,45,115)(27,46,116)(28,47,117)(29,48,118)(30,41,119)(31,42,120)(32,43,113)(33,86,70)(34,87,71)(35,88,72)(36,81,65)(37,82,66)(38,83,67)(39,84,68)(40,85,69)(57,90,109)(58,91,110)(59,92,111)(60,93,112)(61,94,105)(62,95,106)(63,96,107)(64,89,108)(73,102,126)(74,103,127)(75,104,128)(76,97,121)(77,98,122)(78,99,123)(79,100,124)(80,101,125), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(41,68)(42,69)(43,70)(44,71)(45,72)(46,65)(47,66)(48,67)(57,78)(58,79)(59,80)(60,73)(61,74)(62,75)(63,76)(64,77)(81,116)(82,117)(83,118)(84,119)(85,120)(86,113)(87,114)(88,115)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,121)(97,107)(98,108)(99,109)(100,110)(101,111)(102,112)(103,105)(104,106)(129,143)(130,144)(131,137)(132,138)(133,139)(134,140)(135,141)(136,142), (1,36,61)(2,62,37)(3,38,63)(4,64,39)(5,40,57)(6,58,33)(7,34,59)(8,60,35)(9,118,121)(10,122,119)(11,120,123)(12,124,113)(13,114,125)(14,126,115)(15,116,127)(16,128,117)(17,77,30)(18,31,78)(19,79,32)(20,25,80)(21,73,26)(22,27,74)(23,75,28)(24,29,76)(41,130,98)(42,99,131)(43,132,100)(44,101,133)(45,134,102)(46,103,135)(47,136,104)(48,97,129)(49,85,90)(50,91,86)(51,87,92)(52,93,88)(53,81,94)(54,95,82)(55,83,96)(56,89,84)(65,105,141)(66,142,106)(67,107,143)(68,144,108)(69,109,137)(70,138,110)(71,111,139)(72,140,112), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)>;

G:=Group( (1,36,61)(2,37,62)(3,38,63)(4,39,64)(5,40,57)(6,33,58)(7,34,59)(8,35,60)(9,121,118)(10,122,119)(11,123,120)(12,124,113)(13,125,114)(14,126,115)(15,127,116)(16,128,117)(17,77,30)(18,78,31)(19,79,32)(20,80,25)(21,73,26)(22,74,27)(23,75,28)(24,76,29)(41,130,98)(42,131,99)(43,132,100)(44,133,101)(45,134,102)(46,135,103)(47,136,104)(48,129,97)(49,85,90)(50,86,91)(51,87,92)(52,88,93)(53,81,94)(54,82,95)(55,83,96)(56,84,89)(65,105,141)(66,106,142)(67,107,143)(68,108,144)(69,109,137)(70,110,138)(71,111,139)(72,112,140), (1,53,141)(2,54,142)(3,55,143)(4,56,144)(5,49,137)(6,50,138)(7,51,139)(8,52,140)(9,24,129)(10,17,130)(11,18,131)(12,19,132)(13,20,133)(14,21,134)(15,22,135)(16,23,136)(25,44,114)(26,45,115)(27,46,116)(28,47,117)(29,48,118)(30,41,119)(31,42,120)(32,43,113)(33,86,70)(34,87,71)(35,88,72)(36,81,65)(37,82,66)(38,83,67)(39,84,68)(40,85,69)(57,90,109)(58,91,110)(59,92,111)(60,93,112)(61,94,105)(62,95,106)(63,96,107)(64,89,108)(73,102,126)(74,103,127)(75,104,128)(76,97,121)(77,98,122)(78,99,123)(79,100,124)(80,101,125), (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(41,68)(42,69)(43,70)(44,71)(45,72)(46,65)(47,66)(48,67)(57,78)(58,79)(59,80)(60,73)(61,74)(62,75)(63,76)(64,77)(81,116)(82,117)(83,118)(84,119)(85,120)(86,113)(87,114)(88,115)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,121)(97,107)(98,108)(99,109)(100,110)(101,111)(102,112)(103,105)(104,106)(129,143)(130,144)(131,137)(132,138)(133,139)(134,140)(135,141)(136,142), (1,36,61)(2,62,37)(3,38,63)(4,64,39)(5,40,57)(6,58,33)(7,34,59)(8,60,35)(9,118,121)(10,122,119)(11,120,123)(12,124,113)(13,114,125)(14,126,115)(15,116,127)(16,128,117)(17,77,30)(18,31,78)(19,79,32)(20,25,80)(21,73,26)(22,27,74)(23,75,28)(24,29,76)(41,130,98)(42,99,131)(43,132,100)(44,101,133)(45,134,102)(46,103,135)(47,136,104)(48,97,129)(49,85,90)(50,91,86)(51,87,92)(52,93,88)(53,81,94)(54,95,82)(55,83,96)(56,89,84)(65,105,141)(66,142,106)(67,107,143)(68,144,108)(69,109,137)(70,138,110)(71,111,139)(72,140,112), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144) );

G=PermutationGroup([[(1,36,61),(2,37,62),(3,38,63),(4,39,64),(5,40,57),(6,33,58),(7,34,59),(8,35,60),(9,121,118),(10,122,119),(11,123,120),(12,124,113),(13,125,114),(14,126,115),(15,127,116),(16,128,117),(17,77,30),(18,78,31),(19,79,32),(20,80,25),(21,73,26),(22,74,27),(23,75,28),(24,76,29),(41,130,98),(42,131,99),(43,132,100),(44,133,101),(45,134,102),(46,135,103),(47,136,104),(48,129,97),(49,85,90),(50,86,91),(51,87,92),(52,88,93),(53,81,94),(54,82,95),(55,83,96),(56,84,89),(65,105,141),(66,106,142),(67,107,143),(68,108,144),(69,109,137),(70,110,138),(71,111,139),(72,112,140)], [(1,53,141),(2,54,142),(3,55,143),(4,56,144),(5,49,137),(6,50,138),(7,51,139),(8,52,140),(9,24,129),(10,17,130),(11,18,131),(12,19,132),(13,20,133),(14,21,134),(15,22,135),(16,23,136),(25,44,114),(26,45,115),(27,46,116),(28,47,117),(29,48,118),(30,41,119),(31,42,120),(32,43,113),(33,86,70),(34,87,71),(35,88,72),(36,81,65),(37,82,66),(38,83,67),(39,84,68),(40,85,69),(57,90,109),(58,91,110),(59,92,111),(60,93,112),(61,94,105),(62,95,106),(63,96,107),(64,89,108),(73,102,126),(74,103,127),(75,104,128),(76,97,121),(77,98,122),(78,99,123),(79,100,124),(80,101,125)], [(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,55),(10,56),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(31,40),(32,33),(41,68),(42,69),(43,70),(44,71),(45,72),(46,65),(47,66),(48,67),(57,78),(58,79),(59,80),(60,73),(61,74),(62,75),(63,76),(64,77),(81,116),(82,117),(83,118),(84,119),(85,120),(86,113),(87,114),(88,115),(89,122),(90,123),(91,124),(92,125),(93,126),(94,127),(95,128),(96,121),(97,107),(98,108),(99,109),(100,110),(101,111),(102,112),(103,105),(104,106),(129,143),(130,144),(131,137),(132,138),(133,139),(134,140),(135,141),(136,142)], [(1,36,61),(2,62,37),(3,38,63),(4,64,39),(5,40,57),(6,58,33),(7,34,59),(8,60,35),(9,118,121),(10,122,119),(11,120,123),(12,124,113),(13,114,125),(14,126,115),(15,116,127),(16,128,117),(17,77,30),(18,31,78),(19,79,32),(20,25,80),(21,73,26),(22,27,74),(23,75,28),(24,29,76),(41,130,98),(42,99,131),(43,132,100),(44,101,133),(45,134,102),(46,103,135),(47,136,104),(48,97,129),(49,85,90),(50,91,86),(51,87,92),(52,93,88),(53,81,94),(54,95,82),(55,83,96),(56,89,84),(65,105,141),(66,142,106),(67,107,143),(68,144,108),(69,109,137),(70,138,110),(71,111,139),(72,140,112)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144)]])

72 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 6A ··· 6E 6F 6G 6H 6I 6J 6K 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12J 12K ··· 12R 12S 12T 24A ··· 24P order 1 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 6 ··· 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 12 ··· 12 12 ··· 12 12 12 24 ··· 24 size 1 1 9 9 2 ··· 2 4 4 4 4 1 1 9 9 2 ··· 2 4 4 4 4 18 18 3 3 3 3 27 27 27 27 2 ··· 2 4 ··· 4 18 18 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 C8 S3 S3 Dic3 D6 Dic3 C3⋊C8 C4×S3 S3×C8 S32 S3×Dic3 S3×C3⋊C8 kernel C3⋊S3×C3⋊C8 C32×C3⋊C8 C33⋊7C8 C12×C3⋊S3 C3×C3⋊Dic3 C6×C3⋊S3 C3×C3⋊S3 C3×C3⋊C8 C4×C3⋊S3 C3⋊Dic3 C3×C12 C2×C3⋊S3 C3⋊S3 C3×C6 C32 C12 C6 C3 # reps 1 1 1 1 2 2 8 4 1 1 5 1 4 8 16 4 4 8

Matrix representation of C3⋊S3×C3⋊C8 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 1 70 0 0 0 0 1 71 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 52 21 0 0 0 0 59 21 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 10 0 0 0 0 0 0 10 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,1,0,0,0,0,70,71,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[52,59,0,0,0,0,21,21,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,0,10,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3⋊S3×C3⋊C8 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_3\rtimes C_8
% in TeX

G:=Group("C3:S3xC3:C8");
// GroupNames label

G:=SmallGroup(432,431);
// by ID

G=gap.SmallGroup(432,431);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,58,571,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^3=e^8=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽